Q. 16 - Calculus

Question

Find all values of $a$ and $b$ such that $(a, b)$ represents the same point whether it is given in rectangular coordinates or polar coordinates.

Step-by-Step Solution

Answer

The required point is $(0,0)$.

Step 1: Given information

The point $(a, b)$.

Step 2: Calculation

Consider the point $(a, b)$.

The objective is to find a point which is same in polar coordinate system and rectangular coordinate system.

The point which is same in polar coordinate system and rectangular coordinate system is $(0,0)$.

Now suppose $(0,0)$ is a point in rectangular system.

In the coordinate (0,0), x=0 and y=0.

To find the value of $r$ use the equation $r=\sqrt{x^2+y^2}$.

Then,

$$\begin{gathered} r=\sqrt{(0{)}^2+(0{)}^2} [\text{ since }x=0,y=0] \\ r=\sqrt{0} \\ r=0 \end{gathered}$$

To calculate$\theta$ use the formula $\theta ={\tan }^{-1}\left(\frac{y}{x}\right)$.

Then

$$\begin{gathered} \theta ={\tan }^{-1}\left(\frac{0}{0}\right)[sincex=0,y=0] \\ \theta ={\tan }^{-1}\left(0\right) \\ \theta =0 \text{ [since }\tan 2\pi =\tan \pi =0] \end{gathered}$$

Then $(r,\theta )=(0,0)$.

Thus, the coordinate $(0,0)$ is same in polar coordinate system and rectangular coordinate system.

Therefore, the required point is $(0,0).$